In our company we have quite a few people who work only part-time, e.g. some don't work on Wednesdays, some on Tuesdays etc. In addition,

This is possible, and easy :
define new Calendars :
1) off on wednesdays : define each wednesday as off days
2) off on tuesdays : define each tuesday as off days
You'll then just have to assign correct calendar to resources.

some work on specific days only part of the day.

It is not possible to take this cas into account.

Hello
I have the same issue, except that Joe is off every Wednesday afternoon.
-> I tried with FTE = 0.9 (like recommended in manual), but it generates an availability of 0.9 every day and that is not what is expected
-> Calendar (like recommended in this thread) allows to define only full days off
-> I tried with periodic meetings, it generates a lot of meetings and it requires lot of cpu if we want to edit the periodic meetings for a long period (like 1 year). And I could not set the meeting's project to "Administrative" type because in that case it does not appear in the planning. Then it works, but in "real work" interface, the resource can see all planned meetings (instead of only the meeting of the week) (existing ticket #1362 ). So that it not really satisfying.

What is the best solution to handle this ?

Best way to do this is to use Administrative project.
Administrative projects are used to record all the "not available" days (off days, sickness, training, part-time)...
You just define a project and corresponding activities (in your exemple, you can define "part time" activity.).
Then, resource must enter real work every wednesday for 0,5. It is not automatic, but really it will take less than 10mn to enter full year, and the best is to let each resource enter such "unavailable" periods (do not bother project leader with such inputs).

OK, I will experiment this
It is less work than defining periodic meetings anyway
Thanks !